For the potato squash cakes: Preheat the oven to 350 degrees F.
Drizzle the squash halves with oil, sprinkle with salt and pepper and put on a baking sheet cut-side up. Roast until tender, 45 to 60 minutes. Set aside to cool.
Put the potatoes in a pot and add enough salted water to cover. Cover the pot, bring to a boil, lower the heat and simmer until tender, 20 to 25 minutes. Drain the potatoes and mash with a fork or potato masher.
Measure out 1 cup of the squash and 2 cups of the potatoes into a large bowl (save any remaining vegetables for another use). Add the egg yolks and some salt and pepper and blend well. Shape into four equal cakes.
Heat a saute pan over medium-high heat. Add some oil and then the cakes and cook until golden brown on one side. Flip and cook on the second side until golden brown. Keep warm.
For the pork chops and apple gravy: Warm the brown gravy and mustard in a saucepan over medium heat, stirring throughout. Keep on medium-low heat while you cook the chops.
Sprinkle the pork chops on both sides with 1 tablespoon salt and 1 teaspoon pepper. Heat a large pan over medium-high heat until warmed. Ad the oil and the chops and cook until browned on one side, 5 to 6 minutes; flip and cook on the other side another 6 minutes. Remove the chops and keep warm.
Add the apples to the pork chop pan and cook until softened, 3 to 4 minutes. Transfer the apples to the saucepan with the brown sauce and mustard; stir and taste for seasoning and thickness of sauce, adjusting with chicken stock and salt and pepper if needed. Finish by stirring in the butter.
To serve, place a potato squash cake on each plate and then a chop. Finish with the apple gravy and serve.